Post-Newtonian spin and angular momentum of bounded systems

We generalize the newtonian expressions for the orbital angular momentum of a two-body system, and for the spin of each body, by introducing corresponding definitions in the post-Newtonian approximation of fully conservative theories of gravity. Using this definition of the spin and assuming that the bodies rotate rigidly and that the equations of motion are Hamiltonian, we show that in fully conservative theories of gravity the spin of each body undergoes a relativistic precession about the direction of the orbital angular momentum, as a consequence of the local equations of motion for a perfect fluid.Visiting scientist to the Max-Planck-Institut für Physik und Astrophysik. On leave of absence from the Astronomy Department, University of Thessaloniki, Greece.     

Quasi-local gravitational angular momentum and centre of mass from generalised Witten equations

Witten’s proof for the positivity of the ADM mass gives a definition of energy in terms of three-surface spinors. In this paper, we give a generalisation for the remaining six Poincaré charges at spacelike infinity, which are the angular momentum and centre of mass. The construction improves on certain three-surface spinor equations introduced by Shaw. We solve these equations asymptotically obtaining the ten Poincaré charges as integrals over the Nester–Witten two-form. We point out that the defining differential equations can be extended to three-surfaces of arbitrary signature and we study them on the entire boundary of a compact four-dimensional region of spacetime. The resulting quasi-local expressions for energy and angular momentum are integrals over a two-dimensional cross-section of the boundary. For any two consecutive such cross-sections, conservation laws are derived that determine the influx (outflow) of matter and gravitational radiation.     

The analysis of rovibrational motion equations of a diatomic molecule in the linear regime

The motion equations of diatomic molecules are here extended from the absolute vibrational case to a more general and real rotational and vibrational (rovibrational) case. The rovibrational Hamiltonian is heuristically formed by substituting the respective number and angular momentum operators for the vibrational and rotational quantum numbers in the energy eigenvalues of a diatomic molecule which was first introduced by Dunham. The motion equations of observable quantities such as the position and linear momentum are then determined by implementing the well-known Heisenberg relation in quantum mechanics. We face with the second-order imaginary differential equations for describing the temporal variations of the relative position and the linear momentum of two oscillating atoms, which are coupled in the x–y horizontal plane. The possible rovibrational oscillations are distinguished by the three quantum numbers n, l and m associated with the energy and angular momentum quantities. It is finally demonstrated that the simultaneous solutions of rovibrational equations satisfy the energy conservation during all quantised oscillations of a diatomic molecule in space.     

General relativistic dynamics of compact binary systems

The equations of motion of compact binary systems have been derived in the post-Newtonian (PN) approximation of general relativity. The current level of accuracy is 3.5PN order. The conservative part of the equations of motion (neglecting the radiation reaction damping terms) is deducible from a generalized Lagrangian in harmonic coordinates, or equivalently from an ordinary Hamiltonian in ADM coordinates. As an application, we investigate the problem of the dynamical stability of circular binary orbits against gravitational perturbations up to the 3PN order. We find that there is no innermost stable circular orbit or ISCO at the 3PN order for equal masses. To cite this article: L. Blanchet, C. R. Physique 8 (2007).     

Application of energy and angular momentum balance to gravitational radiation reaction for binary systems with spin-orbit coupling

We study gravitational radiation reaction in the equations of motion for binary systems with spin-orbit coupling, at order (v/c)⁷ beyond Newtonian gravity, or O(v/c)² beyond the leading radiation reaction effects for non-spinning bodies. We use expressions for the energy and angular momentum flux at infinity that include spin-orbit corrections, together with an assumption of energy and angular momentum balance, to derive equations of motion that are valid for general orbits and for a class of coordinate gauges. We show that the equations of motion are compatible with those derived earlier by a direct calculation.     

Gravitational interaction of two spinning particles in general relativity

We consider gravitational interaction between two spinning pointlike particles. We use a fastmotion approximation and we obtain the first-order gravitational field and motion equations. Following the method developed by Bel and Martin we get up to the first order: the accelerations, momentum, energy, and a Hamiltonian of the system. This Hamiltonian, when it is expanded in a power series ofc ^–1, agrees with those of earlier authors, who use different techniques.     

线型分子的电子能级之Λ分裂的新解释

认为电子态能级发生Λ分裂虽然是角动量引起的,但不是耦合而是运动牵连作用的结果.从这一认识出发,利用欧拉方程和拉格朗日方程分别得到了包含Λ分裂的分子转动哈密顿函数和哈密顿算符,所得结果与Van.Vleck的著名结论完全一致 关键词： Λ分裂 线型分子 欧拉方程 哈密顿算符     

Mechanism of angular momentum exchange between molecules and Laguerre-Gaussian beams

We derive the interaction Hamiltonian between a diatomic molecule and a Laguerre-Gaussian beam under the assumption of a small spread of the center of mass wave function of the molecule in comparison with the beam waist. Considering the dynamical variables of the center of mass, vibrational, rotational, and electronic motion, we show that, within the electronic dipole approximation, the orbital angular momentum of the field couples with the rotational and electronic motion. The changes in the transition probabilities and selection rules induced by the field orbital angular momentum and the applicability of the derived interaction mechanisms for polyatomic molecules are discussed.     

On the QCD pomeron at large momentum transfer

We investigate the asymptotics of the QCD scattering amplitude with vacuum quantum number exchange in the region where the energy is large and the momentum transfer is much smaller than the energy but large compared to the hadronic scale. The appropriate modification of the QCD pomeron partial wave equation is derived. The singularities in the angular momentum which arise at relatively large momentum transfer are studied in double logarithmic approximation.     

The falling cat as a port-controlled Hamiltonian system

In this article, the falling cat is modeled as two jointed axial symmetric cylinders with arbitrary twist under the constraint of the vanishing total angular momentum. As a control system with the constraint taken into account, this model is formulated as a port-controlled Hamiltonian system defined on the cotangent bundle of the shape space for the jointed cylinders. A control is then designed as a function on the cotangent bundle, according to a standard procedure. Thus, the equations of motion are determined on the cotangent bundle together with the control. The whole motion as a vibrational motion of the falling cat is obtained after integrating the constraint equation of the vanishing total angular momentum. An example of the falling cat is given in which the model turns a somersault to approach a target state in equilibrium with an expected rotation after finishing a vibrational motion.     

The zero-point energy for rotation

The Gaussian overlap approach (GOA) becomes inappropriate for describing the rotation of weakly deformed systems. A modification is proposed which allows to maintain the GOA for small deformations. The zero-point energy subtraction, derived from it, provides a simple and reliable approximation for angular momentum projection. It becomes obvious, however, that the projection complicates the equations which determine the motion along the deformation path. These effects are studied in some simple models and the results are condensed into a simple interpolation formula for the total zero-point energy.     

Role of surface integrals in the Hamiltonian formulation of general relativity

It is shown that if the phase space of general relativity is defined so as to contain the trajectories representing solutions of the equations of motion then, for asymptotically flat spaces, the Hamiltonian does not vanish but its value is given rather by a nonzero surface integral. If the deformations of the surface on which the state is defined are restricted so that the surface moves asymptotically parallel to itself in the time direction, then the surface integral gives directly the energy of the system, prior to fixing the coordinates or solving the constraints. Under more general conditions (when asymptotic Poincaré transformations are allowed) the surface integrals giving the total momentum and angular momentum also contribute to the Hamiltonian. These quantities are also identified without reference to a particular fixation of the coordinates. When coordinate conditions are imposed the associated reduced Hamiltonian is unambiguously obtained by introducing the solutions of the constraints into the surface integral giving the numerical value of the unreduced Hamiltonian. In the present treatment there are therefore no divergences that cease to be divergences after coordinate conditions are imposed. The procedure of reduction of the Hamiltonian is explicity carried out for two cases: (a) Maximal slicing, (b) ADM coordinate conditions.A Hamiltonian formalism which is manifestly covariant under Poincaré transformations at infinity is presented. In such a formalism the ten independent variables describing the asymptotic location of the surface are introduced, together with corresponding conjugate momenta, as new canonical variables in the same footing with the g_ij, π^ij. In this context one may fix the coordinates in the “interior” but still leave open the possibility of making asymptotic Poincaré transformations. In that case all ten generators of the Poincaré group are obtained by inserting the solution of the constraints into corresponding surface integrals.     

Spin force and torque in non-relativistic Dirac oscillator on a sphere

The spin force operator on a non-relativistic Dirac oscillator (in the non-relativistic limit the Dirac oscillator is a spin one-half 3D harmonic oscillator with strong spin–orbit interaction) is derived using the Heisenberg equations of motion and is seen to be formally similar to the force by the electromagnetic field on a moving charged particle. When confined to a sphere of radius R, it is shown that the Hamiltonian of this non-relativistic oscillator can be expressed as a mere kinetic energy operator with an anomalous part. As a result, the power by the spin force and torque operators in this case are seen to vanish. The spin force operator on the sphere is calculated explicitly and its torque is shown to be equal to the rate of change of the kinetic orbital angular momentum operator, again with an anomalous part. This, along with the conservation of the total angular momentum, suggests that the spin force exerts a spin-dependent torque on the kinetic orbital angular momentum operator in order to conserve total angular momentum. The presence of an anomalous spin part in the kinetic orbital angular momentum operator gives rise to an oscillatory behavior similar to the Zitterbewegung. It is suggested that the underlying physics that gives rise to the spin force and the Zitterbewegung is one and the same in NRDO and in systems that manifest spin Hall effect.     

On quadrupole and octupole gravitational radiation in the ANK formalism

Following the approach of Adamo–Newman–Kozameh (ANK) we derive the equations of motion for the center of mass and intrinsic angular moment for isolated sources of gravitational waves in axially symmetric spacetimes. The original ANK formulation is generalized so that the angular momentum coincides with the Komar integral for a rotational Killing symmetry. This is done using the Winicour–Tamburino Linkages which yields the mass dipole-angular momentum tensor for the isolated sources. The ANK formalism then provides a complex worldline in a fiducial flat space to define the notions of center of mass and spin. The equations of motion are derived and then used to analyse a very simple astrophysical process where only quadrupole and octupole contributions are included. The results are then compared with those coming from the post newtonian approximation.     

Discrete gradients in discrete classical mechanics

A simple model of discrete classical mechanics is given where, starting from the continuous Hamilton equations, discrete equations of motion are established together with a proper discrete gradient definition. The conservation laws of the total discrete momentum, angular momentum, and energy are demonstrated.     

Quadrupole test particle as a detector of gravitational waves

In this paper it is shown that in general relativity the theory of motion of quadrupole test particles (QTP's) can be used to describe the energy and angular momentum absorption by detectors of gravitational waves. By specifying the form of the quadrupole moment tensor Taub's [7] equations of motion of QTP's are simplified. In these equations the terms describing the change of the mass and of the angular momentum of a QTP due to external gravitational waves are found to occur. The limiting case of the flat space-time is also briefly discussed.     

Phenomenological description of relativistic-quark interaction on the basis of the Dirac equation with the Cornell potential

Relativistic-quark interaction is described phenomenologically on the basis of the Dirac equation with the Cornell potential. A general form of the initial equation involving the vector and scalar components of the Cornell potential is used for the case of an arbitrary relation between them. The Hamiltonian in the Foldy–Wouthuysen representation is derived in a general form with allowance for electromagnetic interaction. In contrast to earlier investigations, it is relativistic and exact for the zeroth- and first-order terms in the Planck constant and also for those second-order terms that describe contact interactions. General quantum-mechanical equations of motion for the momentum and spin are derived, and the classical limit of the Hamiltonian and for the equations of motion is found for the first time. A relation between the angular velocity of quark spin precession and the force acting on the quark is obtained. The energy of spin–orbit interaction is rather high (on the order of 100 MeV). Terms that describe spin–orbit and contact interactions have opposite signs for the vector and scalar components of the Cornell potential. The evolution of the quark helicity and the spin–spin interaction of the quarks are also calculated.     

Rotating charged black holes in Einstein-Born-Infeld theories and their ADM mass

In this work, the solution of the Einstein equations for a slowly rotating black hole with Born-Infeld charge is obtained. Geometrical properties and horizons of this solution are analyzed. The conditions when the ADM mass (as in the nonlinear static cases) and the ADM angular momentum of the system have been modified by the non linear electromagnetic field of the black hole, are considered.     

Model of the electron spin in stochastic physics

The electron is conceived here as a complex structure composed of a subparticle that is bound to a nearly circular motion. Although in quantum mechanics the spin is not representable, in classical stochastic physics this corresponds to the angular momentum of the subparticle. In fact, assuming Schrödinger-type hydrodynamic equations of motion for the subparticle, the spin-1/2 representation in configuration space and the corresponding Pauli matrices for the electron are obtained. The Hamiltonian of Pauli's theory as the nonrelativistic limit of Dirac's equation is also derived.On sabbatical leave from Universidad de Los Andes, Merida, 5101 Venezuela.     

Double logarithmic asymptotics and regge singularities of quark amplitudes with flavour exchange

We derive simple equations in terms of definite signature partial waves for the quark scattering and annihilation amplitudes in the double logarithmic approximation of QCD. We apply the method of isolating the softest virtual particle in the graphs which is based on gauge invariance and unitarity. Besides the particle Regge trajectories and corresponding cuts, there are further singularities in the angular momentum plane generated by perturbative double logarithmic contributions and which may be of phenomenological relevance. We also consider amplitudes with external currents and discuss how to include ultraviolet single logarithms and non-perturbative contributions.